The bias-variance trade-off that you're referring to has to do with the return estimator. Any RL algorithm you choose needs some estimate of the cumulative return, which is a random variable with many sources of randomness, such as stochastic transitions or rewards.
Monte Carlo RL algorithms estimate returns by running full trajectories and literally averaging the return achieved for each state. This imposes very few assumptions on the system (in fact, you don't even need the Markovian property for these methods), so bias is low. However, variance is high since each estimate depends on the literal trajectories that you observe. As such, you'll need many, many trajectories to get a good estimate of the value function.
On the other hand, with TD methods, you estimate returns as $R_t + \gamma V(S_{t+1})$, where $V$ is your estimate of the value function. Using $V$ this imposes some bias (for instance, the initialization of the value function at the beginning of training affects your next value function estimates), with the benefit of reducing variance. In TD learning, you don't need full environment rollouts to make a return estimate, you just need one transition. This also lets you make much better use of what you've learned about the value function, because you're learning how to infer value "piecewise" rather than just via literal trajectories that you happened to witness.
